### What does Mathematica mean by ComplexInfinity?

*This is my answer to the same question posed on the Mathematics Stack Exchange. It is therefore licenced under CC-BY-SA 3.0.*

# Question

When entered into Wolfram|Alpha, \(\infty^{\infty}\) results in “complex infinity”. Why?

# Answer

WA’s `ComplexInfinity`

is the same as Mathematica’s: it represents a complex “number” which has infinite magnitude but unknown or nonexistent phase.
One can use `DirectedInfinity`

to specify the phase of an infinite quantity, if it approaches infinity in a certain direction.
The standard `Infinity`

is the special case of phase `0`

.
Note that `Infinity`

is different from `Indeterminate`

(which would be the output of e.g. `0/0`

).

Some elucidating examples:

`0/0`

returns`Indeterminate`

, since (for instance) the limit may be approached as \(\frac{1/n}{1/n}\) or \(\frac{2/n}{2/n}\), resulting in two different real numbers.`1/0`

returns`ComplexInfinity`

, since (for instance) the limit may be approached as \(\frac{1}{-1/n}\) or as \(\frac{1}{1/n}\), but every possible way of approaching the limit gives an infinite answer.`Abs[1/0]`

returns`Infinity`

, since the limit is guaranteed to be infinite and approached along the real line in the positive direction.

In your particular example, you get `ComplexInfinity`

because the infinite limit may be approached as (e.g.) \(n^n\) or as \(n^{n+i}\).